# stochasticclock
A module calculating the stochastic deviations in timepoints for atomic clocks.
This module is an application of the theory presented in Galleani et al. (2003), doi:10.1088/0026-1394/40/3/305.
The module's current functionality calculates stochastic deviations using the exact iterative solution to the stochastic differential equation in `Galleani_exact()`
$$\begin{equation*}
\mathbf{X}(t_{n+1}) =
\begin{pmatrix}
1 & \delta t \\\
0 & 1
\end{pmatrix}
\mathbf{X}(t_n) +
\begin{pmatrix}
\delta t \mu_1 + \frac{1}{2} \delta t^2 \mu_2 \\\
\delta t \mu_2
\end{pmatrix}
+ \mathbf{\Sigma}(t_n)
\end{equation*}$$
$$\begin{equation*}
\mathbf{\Sigma}(t_n) \sim \mathcal{N} \bigg( \mathbf{0},
\begin{bmatrix}
\sigma_1^2 \delta t + \frac{1}{3} \sigma_2^2 \delta t^3 & \frac{1}{2}\sigma_2^2 \delta t^2 \\\
\frac{1}{2}\sigma_2^2 \delta t^2 & \sigma_2^2 \delta t
\end{bmatrix}
\bigg)
\end{equation*}$$
Stochastic deviations can be visualised using `clock_error()`, and their distributions simulated with `deviation_distribution()`.
Please consult the Jupyter notebook for a walkthrough of the package.
Raw data
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"description": "# stochasticclock\n\n\n\nA module calculating the stochastic deviations in timepoints for atomic clocks. \n\nThis module is an application of the theory presented in Galleani et al. (2003), doi:10.1088/0026-1394/40/3/305.\n\nThe module's current functionality calculates stochastic deviations using the exact iterative solution to the stochastic differential equation in `Galleani_exact()`\n\n$$\\begin{equation*}\n \\mathbf{X}(t_{n+1}) =\n \\begin{pmatrix}\n 1 & \\delta t \\\\\\ \n 0 & 1 \n \\end{pmatrix}\n \\mathbf{X}(t_n) +\n \\begin{pmatrix}\n \\delta t \\mu_1 + \\frac{1}{2} \\delta t^2 \\mu_2 \\\\\\ \n \\delta t \\mu_2\n \\end{pmatrix} \n + \\mathbf{\\Sigma}(t_n)\n\\end{equation*}$$ \n\n$$\\begin{equation*}\n \\mathbf{\\Sigma}(t_n) \\sim \\mathcal{N} \\bigg( \\mathbf{0},\n \\begin{bmatrix}\n \\sigma_1^2 \\delta t + \\frac{1}{3} \\sigma_2^2 \\delta t^3 & \\frac{1}{2}\\sigma_2^2 \\delta t^2 \\\\\\ \n \\frac{1}{2}\\sigma_2^2 \\delta t^2 & \\sigma_2^2 \\delta t \n \\end{bmatrix} \n \\bigg)\n\\end{equation*}$$ \n\nStochastic deviations can be visualised using `clock_error()`, and their distributions simulated with `deviation_distribution()`.\n\nPlease consult the Jupyter notebook for a walkthrough of the package.\n\n",
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